Bloque III: Documentación de usuario
2. Manual de usuario/admin
Thus, we are interested in adding extra properties to the network or to protocols. In particular, we are interested in ways to restrict the power of the unreliable agents. In Section 4.5, the protocol LNSR was introduced as an adjustment of the LNS protocol. The protocol LNSR is just like LNS with the restriction that unreliable agents are not allowed to initiate calls.
Intuitively, we introduced the protocol LNSR to limit the power of the unreliable agents. Here we show that from the perspective of identification the opposite is true. That is, we show that by executing the protocol LNSR instead of LNS, the chances increase that the unreliable agents remain uncovered. Specifically, we prove that whenever LNSR is successful on a DCU graphG, then so is LNS itself.
Theorem 5.4.15. If LNSR is (strongly) successful in the (strong) complete
sense on a DCU graphG, then so is LNS.
This implies that LNSR⊆LNS, i.e. the class of DCU graphs on which LNSR is (strongly) successful in the (strong) complete sense is a subset of the class of DCU graphs on which LNS is. In particular, we will show that LNSR6= LNS and hence LNSR⊂LNS. Hence restricting the unreliable agents in their communicative power, i.e. not allowing them to initiate any call, shrinks the class of graphs that can be completed.
Proof
The proof of Theorem 5.4.15 is straightforward given that any LNSR-permitted call sequence is also LNS-permitted.
Proof. [Theorem 5.4.15] We have that any LNSR-permitted call sequenceσis also LNS-permitted. So if LNSR is successful in the (strong) complete sense on a DCU graphG, there is an LNS-permitted call sequence σsuch thatGσ is (strong) complete. But then by definition, becauseσ is also LNS-permitted, LNS is also successful onGin the (strong) complete sense.
The reverse direction is not true. There are DCU graphsGon which LNS is successful in the (strong) complete sense, but LNSR is not. There are even DCU graphs Gin which with only terminal unreliable agents for which this holds. This implies that LNSR6= LNS. Consider Examples 5.4.16 and 5.4.17.
Example 5.4.16. Consider the initial DCU network G= (A, R, f, N, S)drawn
below, where A={a, b, c, d},R={a, b, d} and N ={(a, c),(d, b),(d, c)}R. We have that all the unreliable agents are terminal (asA\R={c}=Nc). By trying
out all possible call sequences, we find that LNS is successful (in the complete sense) onG, but LNSR is not.
a b
*U n r e l i a b l e G o s s i p A g e n t T y p e s G o s s i p R e s u l t s U n r e l i a b l e G o s s i p T e s t. Q u i c k C h e c k> s o l v a b l e [(1 ,([1 ,3] , ([ 1] , [] ,[ ]) ,R e l i a b l e, N o n e) ) , (2 ,([2] , ([ 2] , [] , []) ,R e l i a b l e, N o n e) ) , (3 ,([3] , ([ 3] , [] , []) ,A l t e r n a t i n g B l u f f e r,N o n e) ) , (4 ,([2 ,3 ,4] ,([4] ,[] ,[]) ,R e l i a b l e, N o n e) ) ) ] LNS T r u e *U n r e l i a b l e G o s s i p A g e n t T y p e s G o s s i p R e s u l t s U n r e l i a b l e G o s s i p T e s t. Q u i c k C h e c k> s o l v a b l e [(1 ,([1 ,3] , ([ 1] , [] ,[ ]) ,R e l i a b l e, N o n e) ) , (2 ,([2] , ([ 2] , [] , []) ,R e l i a b l e, N o n e) ) , (3 ,([3] , ([ 3] , [] , []) ,A l t e r n a t i n g B l u f f e r,N o n e) ) , (4 ,([2 ,3 ,4] ,([4] ,[] ,[]) ,R e l i a b l e, N o n e) ) ) ] L N S R F a l s e
Example 5.4.17. Consider the initial DCU graphG= (A, R, f, N, S)drawn be-
low, whereA={a, b, c, d},R={b, c, d}andN ={(b, d),(c, a),(c, b),(d, b),(d, c)}R. We have that all the unreliable agents are terminal (asA\R={a}=Na). By
trying out all possible call sequences, we find that LNS is successful in the strong complete sense on G, but LNSR is not. That is, the unreliable agents can be identified when the protocol LNS is executed, but they will remain uncovered when they are limited in their communication.
a b c d *U n r e l i a b l e G o s s i p A g e n t T y p e s G o s s i p R e s u l t s U n r e l i a b l e G o s s i p T e s t. Q u i c k C h e c k> s o l v a b l e S t r o n g [(1 ,([1] , ([ 1] , [] , []) ,A l t e r n a t i n g B l u f f e r,N o n e) ) , (2 ,([2 ,4] , ([ 2] , [] ,[ ]) ,R e l i a b l e, N o n e) ) , (3 ,([1 ,2 ,3] ,([3] ,[] ,[]) ,R e l i a b l e, N o n e) ) , (4 ,([2 ,4] , ([ 4] , [] ,[ ]) ,R e l i a b l e, N o n e) ) ] LNS T r u e *U n r e l i a b l e G o s s i p A g e n t T y p e s G o s s i p R e s u l t s U n r e l i a b l e G o s s i p T e s t. Q u i c k C h e c k> s o l v a b l e S t r o n g [(1 ,([1] , ([ 1] , [] , []) ,A l t e r n a t i n g B l u f f e r,N o n e) ) , (2 ,([2 ,4] , ([ 2] , [] ,[ ]) ,R e l i a b l e, N o n e) ) , (3 ,([1 ,2 ,3] ,([3] ,[] ,[]) ,R e l i a b l e, N o n e) ) , (4 ,([2 ,4] , ([ 4] , [] ,[ ]) ,R e l i a b l e, N o n e) ) ] L N S R F a l s e *U n r e l i a b l e G o s s i p A g e n t T y p e s G o s s i p R e s u l t s U n r e l i a b l e G o s s i p T e s t. Q u i c k C h e c k> s o l v a b l e [(1 ,([1] , ( [1 ] ,[ ] ,[] ) ,A l t e r n a t i n g B l u f f e r,N o n e) ) , (2 ,([2 ,4] , ([ 2] , [] ,[ ]) ,R e l i a b l e, N o n e) ) , (3 ,([1 ,2 ,3] ,([3] ,[] ,[]) ,R e l i a b l e, N o n e) ) , (4 ,([2 ,4] , ([ 4] , [] ,[ ]) ,R e l i a b l e, N o n e) ) ] L N S R F a l s e
Note that in Example 5.4.17, the network can still result in a complete network by executing the protocol LNSR. So indeed LNSR fails in the identification of the unreliable agents.
Consequences and corollaries
In Example 5.4.16, even though the unreliable agents are terminal and are not permitted by LNS to initiate any call in the initial network, they are still required to initiate calls in a later stage of the network in order to obtain strong completeness. In particular, this is the case because initially the network is split into two groups of reliable agents that can only be connected through the unreliable agent. Agentacan only be in touch with agentb and agentdvia the alternating blufferc.
This result is counterintuitive because we introduced LNSR to limit the power of the unreliable agents. But in fact, this can be understood if we think of restricting the communicative power of the unreliable agents as a way of limiting the amount of (contrary) information in the network about the secret of the unreliable agents. If we have an unreliable agentu∈A\R and another agent
a∈R such thata∈Nu butu /∈Na, there will be no communication between them when we execute LNSR.
Yet, it is very interesting that we need the unreliable agents to initiate communication in order to make it easier for the reliable agents to identify them. This raises interesting questions for real-life applications. For instance, it follows from Theorem 5.4.15 that a fake news article will be easier uncovered if this article is actively shared. Or a faulty sensor will be easier identified if its communicative power is not lost due to the error.
As a result of Theorem 5.4.15 and Examples 5.4.16 and 5.4.17, we find that the set of graphs on which LNSR is (strongly) successful in the (strong) complete sense is a proper subset of those graphs on which LNS is. In other words, we have that LNSR⊂LNS. Now, because LNSR is an example of a protocol in which the agents follow different protocols, we can generalize this result to the following proposition.
Conjecture 5.4.18. Consider all DCU graphs with strictly more than one agent.
LetFP be the set of DCU graphs that is solvable (in the (strong) complete sense)
by protocol P and FP0 those that are solvable by protocolP0. ThenFP∩FP0 is a
proper subset of the setFof solvable DCU graphsGin which the agents follow a combination of protocols P and P0 (i.e. A is partitioned into disjoint setsAP
and AP0 such that AP ∪AP0 =A and for each agent a ∈AP we have that a
follows protocolP, and for eacha∈AP0,a followsP0).
Note that FP ∩FP0 ⊆F because we can imagine that some of the agents follow a protocolP whereas some other agents follow the Do Not Call (DNC) protocol, i.e. the protocol that prevents agents from initiating calls. Clearly for DCU graphs with|A|>1 we haveFDN C =∅1. And as such, we have that
1Trivially, whenever we have an initial DCU graph with only one agent, it is solvable by
FP ∩FDN C = FP∩ ∅ =∅. Therefore trivially FP ∩FP0 ⊆F. An example of Conjecture 5.4.18 is the combination of CMO and LNS. We know that LNS is strongly successful (i.e. any LNS-permitted call sequence is successful) on DC graphsGthat are sun graphs [29] and that LNS⊂CMO. Therefore we get the following Conjecture 5.4.19.
Conjecture 5.4.19. The combination of protocols LNS and CMO is strongly
successful on DC graphs Gthat are sun graphs.
Studying combinations of protocols is interesting for real-life applications. There are many networks with no predefined rules for communication. In fact, we may already wonder how agents in an initial network decide on a protocol to execute without outside information—because no communication has yet taken place. In such situations, the chance that agents follow different protocols is very likely. Consider your own network of friends and family. It would be quite an exception if everyone in the network acts according to the same communication protocol.
Discussion
We have shown that any classification of DCU graphs by LNSR is a proper subset of the same classification by LNS. So there are networksGon which the unreliable agents can be identified when they are allowed to initiate calls, but they will remain uncovered when their communication is restricted. This result is very counter-intuitive: unreliability can more easily be detected when it is actively spread. In a sense, this shows that from a point of view of identification, false information that is not actively spread is worse than false information that is.
An implication of this result is that it might be the wrong strategy to cut off or block unreliable agents from the network. Because in that case it might be harder for the rest of the network to identify the unreliable agents as unreliable. One may now wonder whether this is a problem. For unreliable agents that are cut-off from the network cannot harm the network by spreading false information—so why then is it a problem that they are not identified? Clearly there is a more philosophical debate behind this. Yet, it may already be clear that when not all unreliable agents are identified, the agents that are still in touch with the unreliable agent are more likely to believe her.
This raises interesting questions for the applications of our framework, for instance identifying fake news or faulty sensors in a network of electronics. What are now for instance the good measures to take in the battle against fake news? Because from a practical point of view, letting fake news spread actively may not be the desirable solution. Further research is required to investigate these issues and to answer the question how this counter-intuitive consequence weighs against the reasons for preventing false information to spread. We will elaborate on this in Section 6.1.